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No idea specific to that software, but maybe you need to just scroll down through the license agreement (that's a check they sometimes put in to be sure you "read" it all). Once the scrollbar reaches the bottom you may be able to then click Finish.
If not, I'd guess there's either something wrong with the installer (maybe not fully compatible with XPSP3?), or something really weird about your system (compared to what they expect). Since by the time it gets there it's probably checked for compatibility, I really have no idea.
If none of that helps, look for an older installer, or maybe install Eclipse before Aptana if you are allowed to do that. The only experience I can relate is with plugins for graphics programs and sometimes you can just install them to your system as resources to be later used by any graphics app you want, after it's installed. (This only works some of the time, though-- typically it's installed into a specific program directory, one install per app.)

Thanks for the input. I did try scrolling through each license - no dice.

I was able to get around the issue by going back to the previous screen and re-downloading the various packages, where I was then given the option to "Finish" instead of proceeding to the license page. Bizarre.